726 
DE. PLUCKEE ON A NEW GEOMETEY OF SPACE. 
point. This constitution of space is admitted when, in optics, we consider luminous 
points as sending out in all directions luminous rays, or, in mechanics, forces acting on 
points in every direction. 
In the second construction infinite space is likewise regarded as transversed by right 
lines, but these lines are determined by means of planes passing through them. Every 
plane contains an infinite number of right lines having within it every position and 
direction, around each of which the plane may turn. We refer to this second concep- 
tion when, in optics, we regard, instead of rays, the corresponding fronts of waves and 
their consecutive intersections, or when, in mechanics, according to Poinsot’s ingenious 
philosophical views, we introduce into its fundamental principles “ couples,” as well 
entitled to occupy their place as ordinary forces. The instantaneous axes of rotation 
are right lines of the second description. 
3. In order to constitute a new geometry of space, we may fix the position of a right 
line, depending upon four constants, in a different way. We might do it by means of 
four given right lines, by determining, for instance, the shortest distance of any new line 
from each of the four given ones. But all such conceptions were rejected, and the ordi- 
nary system of axes adopted in order to fix the position in space of a right line. Thus 
the new researches, indicated by the foregoing remarks, are intimately connected with 
the usual methods of analytical geometry. The two fragments presented on this occasion 
are only calculated to give an exact idea of the new way of proceeding, and to show its 
importance, greater perhaps than it appears at first sight. 
4. A right line of the first description, which we shall distinguish by the name of ray , 
may be determined by means of two of its projections. We may select the projections 
within the planes XZ and YZ, in order to get, without generalizing, the greatest 
symmetry obtainable, and give to their equations either the form 
cc=rz- J-f,‘ 
y= SZ + <T, 
or 
( 1 ) 
( 2 ) 
tx-\-vjz= 1, 
uy-\-VyZ=. 1 . 
In adopting the first system of equations, the four constants r, s, g>, a are the coordi- 
nates of the ray : two of them, r, s, indicating its direction, the remaining two, §, <r, after 
its direction is determined, giving its position in space. The ray meets the plane XY 
in the point 
%=§, y=°- 
In adopting the second system of equations, we get, in order to determine the same 
ray, the four new constants t, u, v x , v y , which likewise may be regarded as its coordi- 
nates ; t and u ^equal to ^ and ^ indicating the reciprocal values of the intercepts 
cut off on OX and OY by the two projections of the ray, v x and v y ^equal to 
and the reciprocal values of the two intercepts cut off both on OZ. 
